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SPS SPS FM Pure 2025 February Q10

8 marks Challenging +1.2

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where $k$ is a real constant. The planes do not meet at a unique point.

Find the possible values of $k$. [3 marks]
For each value of $k$ found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]

SPS SPS FM Pure 2025 February Q11

8 marks Challenging +1.8

The infinite series $C$ and $S$ are defined by $$C = \cos \theta + \frac{1}{2}\cos 5\theta + \frac{1}{4}\cos 9\theta + \frac{1}{8}\cos 13\theta + \ldots$$ $$S = \sin \theta + \frac{1}{2}\sin 5\theta + \frac{1}{4}\sin 9\theta + \frac{1}{8}\sin 13\theta + \ldots$$ Given that the series $C$ and $S$ are both convergent,

show that $$C + iS = \frac{2e^{i\theta}}{2 - e^{4i\theta}}$$ [4]
Hence show that $$S = \frac{4\sin \theta + 2\sin 3\theta}{5 - 4\cos 4\theta}$$ [4]

SPS SPS FM Pure 2025 February Q12

11 marks Challenging +1.8

The population density $P$, in suitable units, of a certain bacterium at time $t$ hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5. The model uses the differential equation $$\frac{dP}{dt} - \frac{P}{t(1 + t^2)} = \frac{te^{-t}}{\sqrt{1 + t^2}}$$ Find $P$ as a function of $t$. [You may assume that $\lim_{t \to \infty} te^{-t} = 0$]. [11]

SPS SPS FM Pure 2025 February Q13

6 marks Moderate -0.3

Write down the Maclaurin series of $e^x$, in ascending power of $x$, up to and including the term in $x^3$ [1]
Hence, without differentiating, determine the Maclaurin series of $$e^{(x^3-1)}$$ in ascending powers of $x$, up to and including the term in $x^3$, giving each coefficient in simplest form. [5]

SPS SPS FM 2025 October Q1

3 marks Easy -1.2

Determine the equation of the line that passes through the point $(1, 3)$ and is perpendicular to the line with equation $3x + 6y - 5 = 0$. Give your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers to be determined. [3]

SPS SPS FM 2025 October Q2

6 marks Moderate -0.8

In a triangle $ABC$, $AB = 9$ cm, $BC = 7$ cm and $AC = 4$ cm.

Show that $\cos CAB = \frac{2}{3}$. [2]
Hence find the exact value of $\sin CAB$. [2]
Find the exact area of triangle $ABC$. [2]

SPS SPS FM 2025 October Q3

4 marks Moderate -0.5

Given the function $f(x) = 3x^3 - 7x - 1$, defined for all real values of $x$, prove from first principles that $f'(x) = 9x^2 - 7$. [4]

SPS SPS FM 2025 October Q4

8 marks Moderate -0.3

The cubic polynomial $2x^3 - kx^2 + 4x + k$, where $k$ is a constant, is denoted by f(x). It is given that f'(2) = 16.

Show that $k = 3$. [3]

For the remainder of the question, you should use this value of $k$.

Use the factor theorem to show that $(2x + 1)$ is a factor of f(x). [2]
Hence show that the equation f(x) = 0 has only one real root. [3]

SPS SPS FM 2025 October Q5

4 marks Standard +0.3

In this question you must show detailed reasoning. Consider the expansion of $\left(\frac{x^2}{2} + \frac{a}{x}\right)^6$. The constant term is 960. Find the possible values of $a$. [4]

SPS SPS FM 2025 October Q6

6 marks Moderate -0.3

The curve C is defined for $x > 0$ and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$

Find the exact $x$-coordinate of the stationary point giving your answer in the form $a^b$ where $a$ and $b$ are rational numbers. [4]
Find the nature of the stationary point, justifying your answer. [2]

Find the set of values of $x$ for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
Find the equation of C. You may leave your answer in factorised form. [3]

The curve $C_1$ has equation $y = \text{f}(x) + k$, where $k$ is a constant. Given that the graph of $C_1$ intersects the $x$-axis at exactly four places,

find the range of possible values for $k$. [2]

SPS SPS FM 2025 October Q10

4 marks Moderate -0.8

The graph of $y = \text{e}^x$ can be transformed to the graph of $y = \text{e}^{2x-1}$ by a stretch parallel to the $x$-axis followed by a translation.

1. State the scale factor of the stretch. [1]
2. Give full details of the translation. [2]

Alternatively the graph of $y = \text{e}^x$ can be transformed to the graph of $y = \text{e}^{2x-1}$ by a stretch parallel to the $x$-axis and a stretch parallel to the $y$-axis.

State the scale factor of the stretch parallel to the $y$-axis. [1]

SPS SPS FM 2025 October Q11

8 marks Standard +0.3

The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$

Find gf($\text{e}^2$) writing your answer in simplest form. [2]
Find the range of the function fg. [2]
Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form $a \ln 2$ where $a$ is rational. [4]

SPS SPS FM 2025 October Q12

6 marks Standard +0.3

Prove by induction that, for all positive integers $n$, $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]

SPS SPS FM 2025 October Q13

8 marks Challenging +1.8

In this question you must show detailed reasoning. Solve the following equation for $x$ in the interval $0° < x < 180°$ $$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]

SPS SPS SM 2025 October Q1

8 marks Easy -1.8

Express each of the following in the form $px^q$, where $p$ and $q$ are constants.

$\frac{2}{\sqrt[3]{x}}$ [1]
$(5x\sqrt{x})^3$ [1]
$\sqrt{2x^3} \times \sqrt{8x^5}$ [1]
$x^5(27x^6)^{\frac{1}{3}}$ [2]

SPS SPS SM 2025 October Q2

3 marks Moderate -0.3

In this question you must show detailed reasoning. Simplify $10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}$, giving your answer in the form $a + b\sqrt{5}$. [3]

SPS SPS SM 2025 October Q3

5 marks Moderate -0.8

The line $l$ passes through the points $A(-3, 0)$ and $B\left(\frac{5}{3}, 22\right)$

Find the equation of $l$ giving your answer in the form $y = mx + c$ where $m$ and $c$ are constants. [3]

\includegraphics{figure_2} Figure 2 shows the line $l$ and the curve $C$, which intersect at $A$ and $B$. Given that

$C$ has equation $y = 2x^2 + 5x - 3$
the region $R$, shown shaded in Figure 2, is bounded by $l$ and $C$

use inequalities to define $R$. [2]

SPS SPS SM 2025 October Q4

6 marks Moderate -0.8

A sequence has terms $u_1, u_2, u_3, \ldots$ defined by $u_1 = 3$ and $u_{n+1} = u_n^2 - 5$ for $n \geq 1$.
1. Find the values of $u_2$, $u_3$ and $u_4$. [2]
2. Describe the behaviour of the sequence. [1]
The second, third and fourth terms of a geometric progression are 12, 8 and $\frac{16}{3}$. Determine the sum to infinity of this geometric progression. [3]